Commit dcce432f authored by Zdenek Dvorak's avatar Zdenek Dvorak
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Revision of the rest of the paper.

parent 5ada1937
......@@ -79,7 +79,7 @@ A \emph{touching representation by comparable boxes} of a graph $G$ is a touchin
such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable.
Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
Then for a class $\GG$ of graphs, let $\cbdim(\GG):=\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
comparable box dimension of graphs in $\GG$ is not bounded.
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
......@@ -126,21 +126,25 @@ Note that a clique with $2^d$ vertices has a touching representation by comparab
where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$.
Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
In the following we consider the chromatic number $\chi(G)$, and one
of its variants. A \emph{star coloring} of a graph $G$ is a proper
coloring such that any two color classes induce a star forest (i.e., a
graph not containing any 4-vertex path). The \emph{star chromatic
number} $\chi_s(G)$ of $G$ is the minimum number of colors in a star
coloring of $G$. We will need the fact that the star chromatic number
is at most exponential in the comparable box dimension; this follows
In the following we consider the chromatic number $\chi(G)$, and two
of its variants. An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper
coloring such that any two color classes induce a forest (resp. star forest, i.e., a
graph not containing any 4-vertex path). The \emph{acyclic chromatic number} $\chi_a(G)$ (resp. \emph{star chromatic
number} $\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star)
coloring of $G$. We will need the fact that all the variants of the chromatic number
are at most exponential in the comparable box dimension; this follows
from~\cite{subconvex}, although we include an argument to make the
dependence clear.
\begin{lemma}\label{lemma-chrom}
For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot
9^{\cbdim(G)}$.
For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot 9^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
We focus on the star chromatic number and note that the chromatic number may be bounded similarly. Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$ be the vertices of $G$ written so that $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$. Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy colouring $c$ so that $c(i)$ is the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly.
Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$
be the vertices of $G$ written so that $\vol(f(v_1)) \geq \ldots \geq \vol(f(v_n))$.
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy colouring $c$ so that $c(i)$ is
the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there
exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices
$v_j$ such that $j<i$ and there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
......@@ -516,14 +520,12 @@ clique-sums.
\end{itemize}
\end{proof}
The following lemma shows that any graphs has a $C^\star$-clique-sum
extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
\ecbdim(G)$ and for any clique $C^\star$.
The following lemma enables us to pick the root clique at the expense of increasing
the dimension by $\omega(G)$.
\begin{lemma}\label{lem-apex-cs}
For any graph $G$ and any clique $C^\star$, we have that $G$ admits a
$C^\star$-clique-sum extendable touching representation by comparabe
boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus
V(C^\star))$.
For any graph $G$ and any clique $C^\star$, the graph $G$ admits a
$C^\star$-clique-sum extendable touching representation by comparable
boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$.
\end{lemma}
\begin{proof}
The proof is essentially the same as the one of
......@@ -532,40 +534,43 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus
V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct
the desired representation $h$ of $G$ as follows. For each vertex
$v_i\in V(C^\star)$ let $h(v_i)$ be the box fulfilling (v1) with
$d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$, if $i\le
k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] =
$v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined
by the condition (v1) with $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$,
if $i\le k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] =
[1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i] =
\alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values $\alpha_i>0$
are chosen suffciently small so that $h(u)[i] \subset [0,1)$, whenever $u\notin V(C^\star)$.
\alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$
is chosen suffciently small so that $h(u)[i] \subset [0,1)$ whenever $u\notin V(C^\star)$.
We proceed similarly for the clique points. For any
clique $C$ of $G$, if $i\le k$ then let $p(C)[i] = 0$ if $v_i \in
V(C)$, and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we have
to refer to the clique point $p'(C')$ of $C'=C\setminus
\{v_1,\ldots,v_k\}$, as we set $p(C)[i] = \alpha_i p'(C')[i-k]$.
clique $C$ of $G$, if $i\le k$ then let $p(C)[i] = 0$ if $v_i \in V(C)$,
and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus
\{v_1,\ldots,v_k\}$, and we set $p(C)[i] = \alpha p'(C')[i-k]$.
By the construction, it is clear that $h$ is a touching representation of $G$.
As $h'(u) \sqsubset h'(v)$ implies that $h(u) \sqsubset h(v)$, and as
$h(u) \sqsubset h(v_i)$, for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a touching representation by comparable boxes.
By the construction, it is clear that $h$ is a representation of $G$.
For the $C^\star$-clique-sum extendability, it is clear that the (vertices) conditions hold.
For the (cliques) condition (c1), let us first consider two distinct cliques $C_1$ and $C_2$
$h(u) \sqsubset h(v_i)$ for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a representation by comparable boxes.
For the $C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction.
For the \textbf{(cliques)} condition (c1), let us consider distinct cliques $C_1$ and $C_2$
of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^\star)$. If $C'_1 = C'_2$,
there is a vertex $v_i \in V(C_1) \setminus V(C_2)$, and $p(C_1)[i] = 0 \neq 1/4 = p(C_2)[i]$.
Otherwise, if $C'_1 \neq C'_2$, we have that $p'(C'_1) \neq p'(C'_2)$, which leads to
Otherwise, if $C'_1 \neq C'_2$, then $p'(C'_1) \neq p'(C'_2)$, which implies
$p(C_1) \neq p(C_2)$ by construction.
For the (cliques) condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ containing $v$.
In the first dimensions $i \le k$, we always have $h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have
$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$ (as in that case $v$ and $v_i$ are adjacent), and if $v_i \notin V(C)$
we have $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$,
we have that $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case
and we now consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, there is an hyperplane
${\mathcal H}' = \{ p\in \mathbb{R}^{d'}\ |\ p[i] = c\}$ that separates $p'(C')$ and $h'(v)$.
This implies that the following hyperplane
${\mathcal H} = \{ p\in \mathbb{R}^{d}\ |\ p[k+i] = \alpha_{k+i} c\}$ separates $p(C)$ and $h(v)$.
Now we consider a vertex $v_i \in V(C^\star)$, and we note that for any clique $C$ containing $v_i$
For the \textbf{(cliques)} condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and
a clique $C$ of $G$ containing $v$. In the dimensions $i\in\{1,\ldots,k\}$, we always have
$h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$, then
$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent;
and if $v_i \notin V(C)$, then $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$.
By the property (c2) of $h'$,
we have $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$.
Next, let us consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, the condition (c2) for $h'$ implies that $p'(C')$ is disjoint from $h'(v)$,
and thus $p(C)$ is disjoint from $h(v)$.
Finally, we consider a vertex $v_i \in V(C^\star)$. Note that for any clique $C$ containing $v_i$,
we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and $h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$
for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that
$h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
......@@ -578,184 +583,213 @@ of $\cbdim(G)$ and $\chi(G)$.
For any graph $G$, $\ecbdim(G) \le \cbdim(G) + \chi(G)$.
\end{lemma}
\begin{proof}
Let $h$ be a touching representation by comparable boxes of $G$ in
Let $h$ be a touching representation of $G$ by comparable boxes in
$\mathbb{R}^d$, with $d=\cbdim(G)$, and let $c$ be a
$\chi(G)$-coloring of $G$. We start with a slightly modified version
of $h$. We first scale $h$ to fit in $(0,1)^d$, and for a
sufficiently small real $\alpha>0$ we increase each box in $h$, by
sufficiently small real $\alpha>0$ we increase each box in $h$ by
$2\alpha$ in every dimension, that is we replace $h(v)[i] = [a,b]$
by $[a-\varepsilon,b+\varepsilon]$ for each vertex $v$ and dimension
$i$. Furthermore $\alpha$ is chosen sufficiently small, so that no
new intersection was created. The obtained representation $h_1$ is
thus an intersection representation of the same graph $G$ such that,
for every clique $C$ of $G$, the intersection $I_C= \cap_{v\in V(C)
h_1(v)}$ is $d$-dimensional. For any maximal clique $C$ of $G$, let
$p_1(C)$ be a point in the interior of $I_C$.
by $[a-\alpha,b+\alpha]$ for each vertex $v$ and dimension
$i$. We choose $\alpha$ sufficiently small so that the boxes representing
non-adjacent vertices remain disjoint, and thus the resulting representation $h_1$ is
an intersection representation of the same graph $G$. Moreover, observe that
for every clique $C$ of $G$, the intersection $I_C=\bigcap_{v\in V(C)} h_1(v)$ is
a box with non-zero edge lengths. For any clique $C$ of $G$, let
$p_1(C)$ be a point in the interior of $I_C$ different from the points
chosen for all other cliques.
Now we add $\chi(G)$ dimensions to make the representation touching
again, and to ensure some space for the clique boxes
$h^\varepsilon(C)$. Formally we define $h_2$ as follows.
$$h_2(u)[i]=\begin{cases}
h_1(u)[i]&\text{ if $i\le d$}\\
[1/5,3/5]&\text{ if $c(u) < i-d$}\\
[0,2/5]&\text{ if $c(u) = i-d$}\\
[1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\
[0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\
[2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
\end{cases}$$
For any clique $C'$ of $G$, let us denote $c(C')$, the color set
$\{c(u)\ |\ u\in V(C')\}$, and let $C$ be one of the maximal cliques
containing $C'$. We now set
$$p_2(C')[i]=\begin{cases}
For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$.
We now set
$$p_2(C)[i]=\begin{cases}
p_1(C) &\text{ if $i\le d$}\\
2/5 &\text{ if $i-d \in c(C')$}\\%\text{if $\exists u\in V(C)$ with $c(u) = i-d$}\\
2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
1/2 &\text{ otherwise}
\end{cases}
$$
As $h_2$ is an extension of $h_1$, and as for all the extra dimensions $j$ ($j>d$)
we have that $2/5 \in h_2(v)[j]$ for every vertex $v$, we have that $h_2$ is an
intersection representation of $G$. To prove that it is touching consider two adjacent
As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$,
$h_2(v)[j]$ is an interval of length $2/5$ containing the point $2/5$ for every vertex $v$,
we have that $h_2$ is an intersection representation of $G$ by comparable boxes.
To prove that it is touching consider two adjacent
vertices $u$ and $v$ such that $c(u)<c(v)$, and let us note that $h_2(u)[d+c(u)] = [0,2/5]$
and $h_2(v)[d+c(u)] = [2/5,4/5]$. By construction, the boxes of $h_1$ are comparable boxes,
and as $h_2$ is an extension of $h_1$ such that for all the extra dimensions $j$ ($j>d$)
the length of $h_2(v)[j]$ is $2/5$ for every vertex $v$, we have that the boxes in $h_2$ are
comparables boxes.
For the $\emptyset$-clique-sum extendability, the (vertices) conditions clearly hold.
For the (cliques) conditions, let us first note that the points $p_1(C)$, defined for
the maximum cliques, are necessarily distinct. This impies that two cliques $C_1$ and $C_2$,
which clique points $p_2(C_1)$ and $p_2(C_2)$ are based on distinct maximum cliques, necessarily lead to distinct points.
In the case that $C_1$ and $C_2$ belong to some maximal clique $C$, we have that $c(C_1) \neq c(C_2)$
and this implies by construction that $p_2(C_1)$ and $p_2(C_2)$ are distinct. Thus (c1) holds.
By construction of $h_1$, we have that if $h_2^{\varepsilon}(C')[i] \cap h_2(v)[i]$ is non-empty for every $i\le d$,
then we have that $h_2^{\varepsilon}(C')[i] \subset h_2(v)[i]$ for every $i\le d$,
and we have that $v$ belongs to some maximal clique $C$ containing $C'$. If $v\notin V(C')$ note that
$p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)]$, while if $v\in V(C')$ we have that
$h_2^{\varepsilon}(C')[i] \subset [2/5,1/2+\varepsilon] \subset h_2(v)[i]$ for every dimension $i>d$,
except if $c(v)=i-d$, and in that case $h_2(v)[i] \cap h_2^{\varepsilon}(C')[i] =
[0,2/5]\cap[2/5,2/5+\varepsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma.
and $h_2(v)[d+c(u)] = [2/5,4/5]$.
For the $\emptyset$-clique-sum extendability, the \textbf{(vertices)} conditions are void.
For the \textbf{(cliques)} conditions, since $p_1$ is chosen to be injective, the mapping $p_2$
is injective as well, implying that (c1) holds.
Consider now a clique $C$ in $G$ and a vertex $v\in V(G)$. If $c(v)\not\in c(C)$, then
$h_2(v)[c(v)+d]=[0,2/5]$ and $p_2(C)[c(v)+d]=1/2$, implying that $h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset$.
If $c(v)\in c(C)$ but $v\not\in V(C)$, then letting $v'\in V(C)$ be the vertex of color $c(v)$,
we have $vv'\not\in E(G)$, and thus $h_1(v)$ is disjoint from $h_1(v')$. Since $p_1(C)$ is contained
in the interior of $h_1(v')$, it follows that $h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset$.
Finally, suppose that $v\in C$. Since $p_1(C)$ is contained in the interior of $h_1(v)$,
we have $h_2^{\varepsilon}(C)[i] \subset h_2(v)[i]$ for every $i\le d$. For $i>d$ distinct from $d+c(v)$,
we have $p_2^{\varepsilon}(C)[i]\in\{2/5,1/2\}$ and $[2/5,3/5]\subseteq h_2(v)[i]$, and thus
$h_2^{\varepsilon}(C)[i] \subset h_2(v)[i]$. For $i=d+c(v)$, we have $p_2^{\varepsilon}(C)[i]=2/5$
and $h_2(v)[i]=[0,2/5]$, and thus $h_2^{\varepsilon}(C)[i] \cap h_2(v)[i]=\{p_2^{\varepsilon}(C)[i]\}$.
Therefore, (c2) holds.
\end{proof}
Together, the lemmas from this section show that comparable box dimension is almost preserved by
full clique-sums.
\begin{corollary}\label{cor-csum}
Let $\GG$ be a class of graphs of chromatic number at most $k$. If $\GG'$ is the class
of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then
$$\cbdim(\GG')\le \cbdim(\GG) + 2k.$$
\end{corollary}
\begin{proof}
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
Without loss of generality, the labelling of the graphs is chosen so that we first
perform the full clique-sum on $G_1$ and $G_2$, then on the resulting graph and $G_3$, and so on.
Let $C^\star_1=\emptyset$ and for $i=2,\ldots,m$, let $C^\star_i$ be the root clique of $G_i$ on which it is
glued in the full clique-sum operation. By Lemmas~\ref{lem-ecbdim-cbdim} and \ref{lem-apex-cs},
$G_i$ has a $C_i^\star$-clique-sum extendable touching representation by comparable boxes in $\mathbb{R}^d$,
where $d=\cbdim(\GG) + 2k$. Repeatedly applying Lemma~\ref{lem-cs}, we conclude that
$\cbdim(G)\le d$.
\end{proof}
By Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}, this gives the following bounds.
\begin{corollary}\label{cor-csump}
Let $\GG$ be a class of graphs of comparable box dimension at most $d$.
\begin{itemize}
\item The class $\GG'$ of graphs obtained from $\GG$ by repeatedly performing full clique-sums
has comparable box dimension at most $d + 2\cdot 3^d$.
\item The class of graphs obtained from $\GG$ by repeatedly performing clique-sums
has comparable box dimension at most $625^d$.
\end{itemize}
\end{corollary}
\begin{proof}
The former bound directly follows from Corllary~\ref{cor-csum} and the bound on the chromatic number
from Lemma~\ref{lemma-chrom}. For the latter one, we need to bound the star chromatic number of $\GG'$.
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
For $i=1,\ldots, m$, suppose $G_i$ has an acyclic coloring $\varphi_i$ by at most $k$ colors.
Note that the vertices of any clique get pairwise different colors, and thus by permuting the colors,
we can ensure that when we perform the full clique-sum, the vertices that are identified have the same
color. Hence, we can define a coloring $\varphi$ of $G$ such that for each $i$, the restriction of
$\varphi$ to $V(G_i)$ is equal to $\varphi_i$. Let $C$ be the union of any two color classes of $\varphi$.
Then for each $i$, $G_i[C\cap V(G_i)]$ is a forest, and since $G[C]$ is obtained from these graphs
by full clique-sums, $G[C]$ is also a forest. Hence, $\varphi$ is an acyclic coloring of $G$
by at most $k$ colors. By~\cite{albertson2004coloring}, $G$ has a star coloring by at most $2k^2-k$ colors.
Hence, Lemma~\ref{lemma-chrom} implies that $\GG'$ has star chromatic number at most $2\cdot 25^d - 5^d$.
The bound on the comparable box dimension of subgraphs of graphs from $\GG'$ then follows from Lemma~\ref{lemma-subg}.
\end{proof}
\section{The strong product structure and minor-closed classes}
A \emph{$k$-tree} is any graph obtained by repeated full clique-sums on cliques of size $k$ from cliques of size at most $k+1$.
A \emph{$k$-tree-grid} is a strong product of a $k$-tree and a path.
An \emph{extended $k$-tree-grid} is a graph obtained from a $k$-tree-grid by adding at most $k$ apex vertices.
Dujmovi{\'c} et al.~\cite{DJM+} proved the following result.
\begin{theorem}\label{thm-prod}
Any graph $G$ is a subgraph of the strong product of a $k$-tree, a path, and $K_m$, where
Any graph $G$ is a subgraph of the strong product of a $k$-tree-grid and $K_m$, where
\begin{itemize}
\item $k=3$ and $m=3$ if $G$ is planar, and
\item $k=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
\end{itemize}
Moreover, for every $t$, there exists a $k$ such that any
$K_t$-minor-free graph $G$ is a subgraph of a graph obtained from
successive clique-sums of graphs, that are obtained from the strong
product of a path and a $k$-tree, by adding at most $k$ apex vertices.
Moreover, for every $t$, there exists an integer $k$ such that any
$K_t$-minor-free graph $G$ is a subgraph of a graph obtained by repeated clique-sums
from extended $k$-tree-grids.
\end{theorem}
Let us first bound the comparable box dimension of a graph in terms of
its Euler genus. As paths and $m$-cliques admit touching
representations with hypercubes of unit size in $\mathbb{R}^{1}$ and
in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by
Lemma~\ref{lemma-sp} it suffice to bound the comparable box
dimension of $k$-trees.
Lemma~\ref{lemma-sp} it suffices to bound the comparable box
dimension of $k$-trees.
\begin{theorem}\label{thm-ktree}
For any $k$-tree $G$, $\cbdim(G) \le \ecbdim(G) \le k+1$.
\end{theorem}
\begin{proof}
Note that there exists a $k$-tree $G'$ having a $k$-clique $C^\star$
such that $G'\setminus V(C^\star)$ corresponds to $G$. Let us construct
a $C^\star$-clique-sum extendable representation of $G'$ and note that
it induces a $\emptyset$-clique-sum extendable representation of
$G$.
Note that $G'$ can be obtained by starting with a $(k+1)$-clique
containing $C^\star$, and by performing successive full clique-sums of
$K_{k+1}$ on a $K_k$ subclique. By Lemma~\ref{lem-cs}, it suffice to
show that $K_{k+1}$, the $(k+1)$-clique with vertex set $\{v_1,
\ldots, v_{k+1}\}$, has a $(K_{k+1} -\{v_{k+1}\})$-clique-sum
extendable touching representation by hypercubes. Let us define such
touching representation $h$ as follows:
\begin{itemize}
\item $h(v_i)[i] = [-1,0] $ if $i\le k$
\item $h(v_i)[j] = [0,1] $ if $i\le k$ and $i\neq j$
\item $h(v_{k+1})[j] = [0,\frac12]$ for any $j$
\end{itemize}
One can easily check that the (vertices)
conditions are fulfilled. For the (cliques) conditions let us set
Let $H$ be a complete graph with $k+1$ vertices and let $C^\star$ be
a clique of size $k$ in $H$. By Lemma~\ref{lem-cs}, it suffices
to show that $H$ has a $C^\star$-clique-sum extendable touching representation
by hypercubes in $\mathbb{R}^{k+1}$. Let $V(C^\star)=\{v_1,\ldots,v_k\}$.
We construct the representation $h$ so that (v1) holds with $d_{v_i}=i$ for each $i$;
this uniquely determines the hypercubes $h(v_1)$, \ldots, $h(v_k)$.
For the vertex $v_{k+1} \in V(H)\setminus V(C^\star)$, we set $h(v_{k+1})=[0,1/2]^{k+1}$.
This ensures that the \textbf{(vertices)} conditions holds.
For the \textbf{(cliques)} conditions, let us set
the point $p(C)$ for every clique $C$ as follows:
\begin{itemize}
\item $p(C)[i] = 0 $ for every $i\le k$ and if $v_i\in C$
\item $p(C)[i] = \frac14 $ for every $i\le k$ and if $v_i\notin C$
\item $p(C)[i] = 0 $ for every $i\le k$ such that $v_i\in C$
\item $p(C)[i] = \frac14 $ for every $i\le k$ such that $v_i\notin C$
\item $p(C)[k+1] = \frac12 $ if $v_{k+1}\in C$
\item $p(C)[k+1] = \frac34 $ if $v_{k+1}\notin C$
\end{itemize}
By construction, it is clear that $p(C) \in h(v_i)$ if and only if
$v_i\in V(C)$. Let us check the other (cliques) conditions.
By construction, it is clear that for each vertex $v\in V(H)$, $p(C) \in h(v)$ if and only if
$v\in V(C)$.
For any two distinct cliques $C_1$ and $C_2$ the points $p(C_1)$ and
$p(C_2)$ are distinct. Indeed, if $|V(C_1)|\ge |V(C_2)|$ there is a
vertex $v_i\in V(C_1)\setminus V(C_2)$, and this implies that
$p(C_1)[i] < p(C_2)[i]$.
For a vertex $v_i$ and a clique $C$, the boxes $h(v_i)$ and
$h^\varepsilon(C)$ intersect if and only if $v_i\in V(C)$. Indeed, if
$v_i\in V(C)$ then $p(C)\in h(v_i)$ and $p(C)\in h^\varepsilon(C)$, and
if $v_i\notin V(C)$ then $h(v_i)[i] = [-1,0]$ if $i\le k$
(resp. $h(v_i)[i] = [0, \frac12]$ if $i= k+1$) and $h^\varepsilon(C)[i] =
[\frac14,\frac14+\varepsilon]$ (resp. $h^\varepsilon(C)[i] =
[\frac34,\frac34+\varepsilon]$). Finally, if $v_i\in V(C)$ we have that
$h(v_i)[i] \cap h^\varepsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j]
\cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for any $j\neq i$
and any $\varepsilon <\frac14$. This concludes the proof of the theorem.
$p(C_2)$ are distinct. Indeed, by symmetry we can assume that for some $i$,
we have $v_i\in V(C_1)\setminus V(C_2)$, and this implies that $p(C_1)[i] < p(C_2)[i]$.
Hence, the condition (v1) holds.
Consider now a vertex $v_i$ and a clique $C$. As we observed before, if $v_i\not\in V(C)$,
then $p(C) \not\in h(v_i)$, and thus $h^\varepsilon(C)$ and $h(v_i)$ are disjoint (for sufficiently small $\varepsilon>0$).
If $v_i\in C$, then the definitions ensure that $p(C)[i]$ is equal to the maximum of $h(v_i)[i]$,
and that for $j\neq i$, $p(C)[j]$ is in the interior of $h(v_i)[j]$, implying
$h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for sufficiently small $\varepsilon>0$.
\end{proof}
The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.
Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree}
extends to graphs of treewidth at most $k$.
\begin{corollary}\label{cor-tw}
Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
\end{corollary}
\begin{proof}
Let $k=\tw(G)$. Observe that there exists a $k$-tree $T$ with the root clique $C^\star$ such that $G\subseteq T-V(C^\star)$.
Inspection of the proof of Theorem~\ref{thm-ktree} (and Lemma~\ref{lem-cs}) shows that we obtain
a representation $h$ of $T-V(C^\star)$ in $\mathbb{R}^{k+1}$ such that
\begin{itemize}
\item the vertices are represented by hypercubes of pairwise different sizes,
\item if $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$, then $h(u)\cap h(v)$ is a facet of $h(u)$ incident
with its point with minimum coordinates, and
\item for each vertex $u$ and each facet of $h(u)$ incident with its point with minimum coordinates, there exists
at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.
\end{itemize}
If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,
we now alter the representation by shrinking $h(u)$ slighly away from $h(v)$ (so that all other touchings are preserved).
Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.
\end{proof}
Note that actually the bound on the comparable boxes dimension of Theorem~\ref{thm-ktree}
extends to graphs of treewidth $k$. For this, note that the construction in this proof can
provide us with a representation $h$ of any $k$-tree $G$ with hypercubes of distinct sizes.
Note also that this representation is such that for any two adjacent vertices $u$ and $v$,
with $h(u) \sqsubset h(v)$ say, the intersection $I = h(u) \cap h(v)$ is a facet of $h(u)$.
Actually $I[i] = h(u)[i]$ for every dimension, except one that we denote $j$. For this
dimension we have that $I[j]=\{c\}$ for some $c$, and that $h(u)[j]=[c,c+s]$,
where $s$ is the length of the sides of $h(u)$. In that context to delete an edge $uv$
one can simply replace $h(u)[j]=[c,c+s]$ with $[c+\varepsilon,c+s]$, for a sufficiently small $\varepsilon$.
One can proceed similarly for any subset of edges, and note that as the hypercubes in $h$ have
distinct sizes these small perturbations give rise to boxes that are still comparable.
Thus for any treewidth $k$ graph $H$ (that is a subgraph of a $k$-tree $G$) we have $\cbdim(H)\le k+1$.
As every planar graph $G$ has a touching representation by cubes in
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le
3$. For the graphs with higher Euler genus we can also derive upper
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le 3$.
For the graphs with higher Euler genus we can also derive upper
bounds. Indeed, combining the previous observation on the
representations of paths and $K_m$, with Theorem~\ref{thm-ktree},
Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain:
\begin{corollary}\label{cor-genus}
For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$
of $G$ such that $\cbdim(G')\le 5+1+\lceil \log_2 \max(2g,3)\rceil$.
Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2
81}.$$
of $G$ such that $\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$.
Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$$
\end{corollary}
Let us now finally prove Theorem~\ref{thm-minor}, using the structure
provided by Theorem~\ref{thm-prod}. We have seen that the strong
product of a path and a $k$-tree has bounded comparable boxes
dimension, and by Lemma~\ref{lemma-apex} adding at most $k$ apex
vertices keeps the dimension bounded. Then by Lemma~\ref{lemma-chrom}
and Lemma~\ref{lem-ecbdim-cbdim}, these graphs admit a
$\emptyset$-clique-sum extendable representations in bounded
dimensions. As the obtained graphs have bounded dimension, by
Lemma~\ref{lemma-cliq} and Lemma~\ref{lem-apex-cs}, for any choice of
a root clique $C^\star$, they have a $C^\star$-clique-sum extendable
representation in bounded dimension. Thus by Lemma~\ref{lem-cs} any
sequence of clique sum from these graphs leads to a graph with bounded
dimension. Finally, we have seen that taking a subgraph does not lead
to undounded dimension, and we obtain Theorem~\ref{thm-minor}.
Similarly, we can deal with proper minor-closed classes.
\begin{proof}[Proof of Theorem~\ref{thm-minor}]
Let $\GG$ be a proper minor-closed class. Since $\GG$ is proper, there exists $t$ such that $K_t\not\in \GG$.
By Theorem~\ref{thm-prod}, there exists $k$ such that every graph in $\GG$ is a subgraph of a graph obtained by repeated clique-sums
from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$,
and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 625^{2k+2}$.
\end{proof}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
and comparable box dimension $n$. It follows that the dependence of the comparable box dimension cannot be
and comparable box dimension $n$. It follows that the dependence of the comparable box dimension on the Euler genus cannot be
subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-genus}
certainly can be improved). The dependence of the comparable box dimension on the size of the forbidden minor that we
established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
......@@ -808,9 +842,7 @@ the smallest axis-aligned hypercube containing $f(v)$, then there exists a posit
$(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$.
\end{itemize}
\note{TO REMOVE if we reintroduce tree decomposition earlier !}
\note{Let us recall some notions about treewidth.
Let us recall some notions about treewidth.
A \emph{tree decomposition} of a graph $G$ is a pair
$(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$
assigns a \emph{bag} to each of its nodes, such that
......@@ -821,11 +853,9 @@ $(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R
non-empty and induces a connected subtree of $T$.
\end{itemize}
For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$.
%For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is nearest to the root of $T$.
%The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$,
%and its
The \emph{width} of the tree decomposition is the maximum of the sizes of the bags minus $1$. The \emph{treewidth} of a graph is the minimum
of the widths of its tree decompositions.}
of the widths of its tree decompositions. Let us remark that the value of treewidth obtained via this definition coincides
with the one via $k$-trees which we used in the previous section.
\begin{theorem}\label{thm-twfrag}
For positive integers $t$, $s$, and $d$, the class of graphs
......@@ -872,7 +902,7 @@ $$\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_
By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$.
Let us now bound the treewidth of $G-X$.
For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus (\cup_{H\in \HH^a} H)$.
For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus \bigl(\bigcup_{H\in \HH^a} H\bigr)$.
A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge 0$.
A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.
Note that there exists a rooted tree $T$ whose vertices are
......
......@@ -5350,3 +5350,11 @@ note = {In Press}
year=1994,
pages={133--138}}
}
@article{albertson2004coloring,
title={Coloring with no $2 $-Colored $ P\_4 $'s},
author={Albertson, Michael O and Chappell, Glenn G and Kierstead, Hal A and K{\"u}ndgen, Andr{\'e} and Ramamurthi, Radhika},
journal={the electronic journal of combinatorics},
pages={R26--R26},
year={2004}
}
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